Question

If for a real number $y,\left[y\right]$ is the greatest integer less than or equal to $y$, then value of the integral $\underset{\frac{\pi }{2}}{\overset{\frac{3\pi }{2}}{\int }}\left[2\sin x\right]dx$, is

• A. $-\pi$
• B. $0$
• C. $-\frac{\pi }{2}$
• D. $\frac{\pi }{2}$

Answer 1

The correct option is C $-\frac{\pi }{2}$

$I=\underset{\frac{\pi }{2}}{\overset{\frac{5\pi }{6}}{\int }}\left(1\right)dx+\underset{\frac{5\pi }{6}}{\overset{\pi }{\int }}0.dx+\underset{\pi }{\overset{\frac{7\pi }{6}}{\int }}(-1)dx+\underset{\frac{7\pi }{6}}{\overset{\frac{3\pi }{2}}{\int }}(-2)dx$
$=(\frac{5\pi }{6}-\frac{\pi }{2})+0-(\frac{7\pi }{6}-\pi )-2(\frac{3\pi }{2}-\frac{7\pi }{6})$
$I=-\frac{\pi }{2}$